Probing interlayer shear thermal deformation in atomically-thin van der Waals layered materials

Atomically-thin van der Waals layered materials, with both high in-plane stiffness and bending flexibility, offer a unique platform for thermomechanical engineering. However, the lack of effective characterization techniques hinders the development of this research topic. Here, we develop a direct experimental method and effective theoretical model to study the mechanical, thermal, and interlayer properties of van der Waals materials. This is accomplished by using a carefully designed WSe2-based heterostructure, where monolayer WSe2 serves as an in-situ strain meter. Combining experimental results and theoretical modelling, we are able to resolve the shear deformation and interlayer shear thermal deformation of each individual layer quantitatively in van der Waals materials. Our approach also provides important interlayer coupling information as well as key thermal parameters. The model can be applied to van der Waals materials with different layer numbers and various boundary conditions for both thermally-induced and mechanically-induced deformations.

Based on the ISTD model, when 2 -1 n N   , ( ) n  satisfies the following equation when temperature changes from T0 to T1: Here, cp is the interlayer coupling coefficient between phosphorene and phosphorene layers. γp and τp are Young's modulus and thermal-induced intrinsic deformation of phosphorene, respectively. τp is a constant depending on the TEC of phosphorene.
When n = 1, the STD of the bottom phosphorene is negligible from T0 to T1 due to the strong clamping effect and small TEC of SiO2 substrates, which gives: (1) 0 When n = N, the mechanical behaviors of the top phosphorene layers are totally different in phosphorene/SiO2 and WSe2/phosphorene/SiO2 systems. For phosphorene/SiO2, the N-th phosphorene merely interacts with the (N-1)-th phosphorene. Therefore, STD of the top layer, τ (N), satisfies the following equation at Where ∆τ(N) is the ISTD between the N-th and the (N-1)-th phosphorene.
Here, ch is the interlayer coupling coefficient between WSe2 and phosphorene layers.

Supplementary Note 4
A local band average approach under Debye approximation has declared that the temperature-dependent thermal expansion coefficient (TEC) is proportional to the specific heat (Cv) and can be expressed as 3 : Where A is a constant. Then the thermally-induced intrinsic strain of the monolayer WSe2 can be obtained by integrating TEC: In this work, 300 K is taken as the initial temperature T0 and 10 K as the final temperature T1. According to previous reports, the Debye temperature of atomically thin WSe2 is around 170 K 4 and the TEC at 300 K 5-8 is around 7×10 -6 K -1 . Utilizing Supplementary Equation (6) and (7), the thermally-induced intrinsic deformation of WSe2, τ WSe2 , can be deduced as -0.17% from 300 to 10 K.

Supplementary Note 5
We have performed experiments to investigate the temperature dependence of the strain gauge factor (for monolayer WSe2. In order to measure the strain gauge factor, we exfoliate monolayer WSe2 directly onto the polyimide (PI) membrane covered by a 50 nm-thick gold film ( Supplementary Fig. 3a). We choose PI instead of PET membrane as the flexible substrate because PI can function well at low temperatures. Here, the 50 nm-thick gold film is sputtered onto PI membrane to eliminate the strong PL background from PI. Then, the WSe2/Au/PI sample is loaded on a home-made strain setup with a two-point bending geometry, as shown in Supplementary Fig. 3b. Through pushing the side screw, the slider will move forward and PI membrane will be bend. Hence, we can only provide the data above 200 K and we sincerely apologize for this.
We plot the photon energy of WSe2 as a function of tensile strain (0.23%, 0.45%, 0.61%, 0.77%) at 200, 250, and 300 K, respectively (scatters in Supplementary Fig. 3c). The fitting results show clear linear dependence between photon energy and strain (dotted line in Supplementary Fig. 3c). The slope is extracted as -0.020, -0.023 and -0.021, based on which the biaxial strain gauge factor can be estimated as -40, -46 and -42 meV/% at 200, 250 and 300 K, respectively. Hence, this experiment demonstrates that the strain gauge factor of WSe2 shows weak temperature dependence, at least, in the range from 200 to 300 K.
Ref. 13 Ref. 14 Ref. 15 Ref. 16 This work Uniaxial (meV/%) Here, we list the strain gauge factor of WSe2 obtained from our experiment and previous reports at room temperature in Supplementary Table 1. It is clear that our experimental results are smaller than previously reported values, which could be attributed to the inefficient strain transfer at the WSe2/Au interface. The real strain gauge factor is always underestimated in experiments. Therefore, in the manuscript, the biaxial strain gauge factor is adopted as -100 meV/%, which stands between the experimental and theoretical values. To further study the influence of strain gauge factor to the fitting results, the strain gauge factor is set to be -80, -100 and -120 meV/%. As shown in Supplementary  (solid circle) layers extracted from our model.